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Chapter 5: Q24E (page 271)

In Problems 19–24, convert the given second-order equation into a first-order system by setting v=y’. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).

y''(t)+y(t)-y(t)3=0

Short Answer

Expert verified

The critical points are (0, 0), (1,0),(-1,0).

Step by step solution

01

Find the critical point

Here the equation is y''(t)+y(t)-y(t)3=0.

Put v=y'  and  v'=y''.

Then the system is;

y'=vy''=-y+y3v'=-y+y3

For critical points equate the system equal to zero.

v=0-y+y3=0y(-1+y2)=0

If y≠0then:

-1+y2=0y=±1

So, the critical point is (0,0)and (1,0)(-1,0).

02

Sketch

This is the required result.

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Most popular questions from this chapter

Use the Runge–Kutta algorithm for systems with h= 0.1 to approximate the solution to the initial value problem.

x'=yz;x(0)=0,y'=-xz;y(0)=1,z'=-xy2;z(0)=1,

At t=1.

Find the critical points and solve the related phase plane differential equation for the system dxdt=(x-1)(y-1),dydt=y(y-1).Describe (without using computer software) the asymptotic behavior of trajectories (as role="math" localid="1664302603010" t→∞) that start at

  1. (3, 2)
  2. (2, 1/2)
  3. ( -2, 1/2)
  4. (3, -2)

In Problems 25 – 28, use the elimination method to find a general solution for the given system of three equations in the three unknown functions x(t), y(t), and z(t).

x'=x+2y+z,y'=6x-y,z'=-x-2y-z

Falling Object. The motion of an object moving vertically through the air is governed by the equation\(\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - g - }}\frac{{\bf{g}}}{{{{\bf{V}}^{\bf{2}}}}}\frac{{{\bf{dy}}}}{{{\bf{dt}}}}\left| {\frac{{{\bf{dy}}}}{{{\bf{dt}}}}} \right|\)where y is the upward vertical displacement and V is a constant called the terminal speed. Take \({\bf{g = 32ft/se}}{{\bf{c}}^{\bf{2}}}\)and V = 50 ft/sec. Sketch trajectories in the yv-phase plane for \( - 100 \le {\bf{y}} \le 100, - 100 \le {\bf{v}} \le 100\)starting from y = 0 and y = -75, -50, -25, 0, 25, 50, and 75 ft/sec. Interpret the trajectories physically; why is V called the terminal speed?

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

y4(t)-y3(t)+7y(t)=cost;y(0)=y'(0)=1,y''(0)=0,y3(0)=2
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